Math 125
Carter
Test 2 Spring 2003
General: Do all your work and write your anwsers inside the blue book. Do not write on
the test. Write your name on only the outside of the blue book. Good luck!
1. For each of the following compute the derivatve of y with respect to x.
dy
(Same things — different notation). Do not
That is find y ′ , f ′(x), or dx
simplify your answer! ( 6 points each)
(a)
(b)
(c)
(d)
(e)
f (x) = x2 − 3x + 2
y=
x
x2 −1
y = (x3 + 3x − 4)−5
f (x) = ex cos (x)
f (x) = x2 cos (x − π2 )
(f)
f (x) = cos (x3 + 3x + 2)
(g)
cos (x) + sin (y) = 1
(h)
y=e
(i)
(j)
√
x
cos (ln (x))
q
f (x) = (1 − x2)
q
y = ln (x2 + 1)
(k)
x2 + y 2 = 1
(l)
f (x) = 2x
3
−4
2. (8 points) Use similar triangles, the squeeze theorem for limits, and
the fact the the area of a circular sector subtended by an angle θ is
2
A(θ) = R2 θ where R is the radius to show that
lim
h↓0
sin (θ)
=1
θ
1
3. (10 points) An anvil thrown upward from the edge of a 496 foot cliff
moves vertically along a straight line according to the equation,
s(t) = −16t2 + 80t + 496
where t ≥ 0 is measured in seconds, and the vertical position, s, is
measured in feet.
(a) Sketch a graph of the velocity as a function of time. Include an
appropriate domain.
(b) When does the anvil reach its acme (highest point)?
(c) What is the velocity of the anvil as it hits the ground (s(t) = 0)?
(d) When does the anvil pass the edge of the cliff?
(e) What happens to the Coyote?
4. (10 points) The diagram below indicates a 15 foot ladder leaning against
a wall. The bottom of the ladder slides away from the wall. Find the
rate of change of the distance from the wall with respect to the angle θ
when the angle is π/3.
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15 ft
y
Find
dy
dθ
θ= π
3
2